导电薄层下三维岩脉的CSAMT响应分析

为了研究探测3D导电岩脉状目标体的可控源音频电磁法（CSAMT）的最佳观测装置,我们用压缩积分方程法数值模拟了赋存于导电薄层下的3D导电岩脉的电磁场响应.计算结果表明,最有效的源的位置是把发射源沿岩脉的走向放置.相对于其他几种装置,这种观测装置得到的异常场与主场的比值最大.利用这种最佳的观测装置,通过将异常场E_ya 和 H_xa与噪音作对比,研究了当岩脉的埋深和高度发生变化时岩脉的可探测性问题.当岩脉的顶部埋深增加时,导电岩脉的可探测性降低.岩脉高度的变化对异常场幅值的影响要小于岩脉顶部埋深所产生的影响.所以通常情况下,即便应用了最有效的CSAMT装置,也很难探测到岩脉的高度,应用钻孔中的探测可能会解决这个问题.     

NUMERICAL MODELLING OF A NEW MULTI-FREQUENCY ELECTROMAGNETIC SYSTEM*

The amplitude of the horizontal magnetic field in the ground between two parallel wires, both carrying an alternating current in the same direction, is likely to have a saddle point if the separation between the wires is small and the frequency is low. The amplitude has a maximum in the vertical direction and a minimum in the horizontal. Rectangular geological structures in the ground which are centered between the wires have a varying effect on the magnetic fields at the surface. In general, the vertical magnetic field “crosses over” at the center of the structure. A shallow and flat lying conductor displays a broad flat type of profile when the horizontal magnetic field between the wires is measured. Changing the structure to a narrower but more conducting one at depth will provide a more pointed but still broad profile. The phase of the horizontal field is also increased. When the structure is a thin vertical dyke, the amplitude of the horizontal magnetic field anomaly due to the dyke rapidly decreases as the depth of the dyke is increased. The phase of the horizontal field is less sensitive to changes in depth of the dyke but is more sensitive to the conductivity ratio of the dyke and the half-space. The amplitude of the vertical magnetic field anomaly due to the dyke is only slightly influenced by conductivity contrast or the depth of the dyke. The phase of the vertical magnetic field, however, is strongly influenced by the conductivity contrast, particularly if the conductivity frequency product is greater than hundred. In essence, the field behaves like that of the conventional vertical loop source, but the fields are uniform over much larger areas. This suggests the possibility of using dip angle measurements for rapid reconnaissance.     

Variation of magnetic properties in sedimentary rocks hosting the Foum Zguid dyke (southern Morocco): Combined effects of re-crystallization and Fe-metasomatism

The effects of dyke intrusion on the magnetic properties of host sedimentary rocks are still poorly understood. Therefore, we have evaluated bulk magnetic parameters of standard palaeomagnetic samples collected along several sections across the sediments hosting the Foum Zguid dyke in southern Morocco. The study has been completed with the evaluation of the magnetic fabric after laboratory application of sequential heating experiments.The present study shows that: (1) close to Foum Zguid dykes, the variations of the bulk magnetic parameters and of the magnetic fabric is strongly related with re-crystallization and Fe-metasomatism intensity. (2) The thermal experiments on AMS of samples collected farther from the dyke and, thus, less affected by heating during dyke emplacement, indicate that 300–400 °C is the minimum experimental temperature necessary to trigger appreciable transformations of the pre-existing magnetic fabrics. For temperatures higher than ca. 580 °C, the magnetic fabric transformations are fully realized, with complete transposition of the initial fabric to a fabric similar to that of samples collected close to the dyke. Therefore, measured variations of the magnetic fabric can be used to evaluate re-crystallization temperatures experienced by the host sedimentary rock during dyke emplacement. The distinct magnetic behaviour observed along the cross-sections strongly suggests that samples collected farther from the dyke margins did not experience thermal episodes with temperatures higher than 300 °C after dyke emplacement. (3) AMS data shows a gradual variation of the magnetic fabric with distance from the dyke margin, from sub-horizontal K₃ away from the dyke to vertical K₃ close to the dyke. Experimental heating shows that heat alone can be responsible for this strong variation. Therefore, such orientation changes should not be unequivocally interpreted as the result of a stress field (resulting from the emplacement of the dyke, for instance). (4) Magnetic studies prove to be a very sensitive tool to assess rock magnetic transformations, thermally and chemically induced by dyke intrusion in hosting sediments.     

The long-term growth of volcanic edifices: numerical modelling of the role of dyke intrusion and lava-flow emplacement

The contribution of intrusive complexes to volcano growth is attested by field observations and by the monitoring of active volcanoes. We used numerical simulations to quantitatively estimate the relative contributions to volcano growth of elastic dislocations related to dyke intrusions and of the accumulation of lava flows. The ground uplift induced by dyke intrusions was calculated with the equations of Okada (Bull. Seismol. Soc. Am., 75 (1985) 1135). The spreading of lava flows was simulated as the flow of a Bingham fluid.With realistic parameters for dyke statistics and lava-flow rheology we find the contribution of dyke intrusions to the growth of a basaltic shield archetype to be about 13% in terms of volume and 30% in terms of height. The result is strongly dependent on the proportion of dykes reaching the surface to feed a lava flow. Systematic testing of the model indicates that edifices tend to be high and steep if dykes are thick and high, issued from a small and shallow magma chamber, and if they feed lava flows of high yield strength.The simulation was applied to Ko'olau (O'ahu Is., Hawai'i) and Piton de la Fournaise (Réunion Is.) volcanoes. The simulation of Ko'olau with dyke parameters as described by Walker (Geology, 14 (1986) 310; U.S. Geol. Surv. Prof. Pap., 1350 (1987) 961) and with lava-flow characteristics collected at Kilauea volcano (Hawai'i Is.) results in an edifice morphology very close to that of the real volcano. The best fit model of the Piton de la Fournaise central cone, with its steep slope and E–W elongation, is obtained by the intrusion of 10 000 short and thick dykes issued from a very small and shallow magma chamber and feeding only 700 low-volume lava flows. The same method may be applied to the growth of basaltic shields and other volcano types in different environments, including non-terrestrial volcanism.     

Local site effects on a high-pier railway bridge under tridirectional spatial excitations: Nonstationary stochastic analysis

This paper presents a theoretical nonstationary stochastic analysis scheme using pseudo-excitation method (PEM) for seismic analysis of long-span structures under tridirectional spatially varying ground motions, based on which the local site effects on structural seismic response are studied for a high-pier railway bridge. An absolute-response-oriented scheme of PEM in nonstationary stochastic analysis of structure under tridirectional spatial seismic motions, in conjunction with the derived mathematical scheme in modeling tridirectional nonstationary spatially correlated ground motions, is proposed to resolve the drawbacks of conventional indirect approach. To apply the proposed theoretical approach readily in stochastic seismic analysis of complex and significant structures, this scheme is implemented and verified in a general finite element platform, and is then applied to a high-pier railway bridge under spatially varying ground motions considering the local site effect and the effect of ground motion nonstationarity. Conclusions are drawn and can be applied in the actual seismic design and analysis of high-pier railway bridges under tridirectional nonstationary multiple excitations.     

嵊泗岛辉绿岩墙群的侵位方式:来自磁组构的证据

岩墙磁组构能反映岩浆的侵位方式.中国东部嵊泗岛广泛发育了晚白垩世辉绿岩岩墙群.我们对其中8条不同走向岩墙进行了采样,沿岩墙两边部及横跨岩墙剖面获得共273个独立定向岩芯样品.岩石磁学分析表明辉绿岩的主要携磁矿物为多畴贫钛磁铁矿,可能含少量磁赤铁矿.各条岩墙的磁组构均具有低的各向异性度Pj＜1.2,且主轴的空间方位各不相...     

Mechanisms for ground-surface fracturing and incipient slope failure associated with the 2001 eruption of Mt. Etna, Italy: analysis of ephemeral field data

During the July–August 2001 eruption of Mt. Etna development of extensional fractures/faults and grabens accompanied magma intrusion and subsequent volcanic activity. During the first days of the eruption, we performed an analysis of attitude, displacement and propagation of fractures and faults exposed on the ground surface in two sites, Torre del Filosofo and Valle del Leone, located along the same fracture system in the region surrounding the Valle del Bove depression on the eastern flank of Mt. Etna. Fractures and faults formed as the consequence of a shallow intruding dyke system that fed the several volcanic centres developed along the fracture system. The investigated sites differ in slope attitude and in geometrical relationships between fractures and slopes. In particular, the fracture system propagated parallel to the gentle slope (<7° dip) in the Torre del Filosofo area, and perpendicular to the steep slope (25° dip) in the Valle del Leone area. In the Torre del Filosofo area, slight graben subsidence and horizontal extension of the ground surface by about 3 m were recorded. In the Valle del Leone area, extensional faulting forming a larger and deeper graben with horizontal extension of the ground surface by about 10 m was recorded. For the Valle del Leone area, we assessed a downhill dip of 14° for the graben master fault at the structural level beneath the graben where the fault dip shallows. These results suggest that dyke intrusion at Mount Etna, and particularly in the region surrounding the Valle del Bove depression, may be at the origin of slope failure and subsequent slumps where boundary conditions, i.e. geometry of dyke, slope dip and initial shear stress, amongst others, favour incipient failures.     

On the use of the autocorrelation method for estimating the depth to the magnetic basement

The depth to the top of magnetic dykes can be estimated from total field aeromagnetic data using the relation between the depth to magnetic sources and the autocorrelation function of magnetic data. By using synthetic anomalies we show that in the ideal case, depth can be determined to an accuracy of 10% or better, when the anomaly sources are two-dimensional dykes. However, the estimated depths depend on the width of the dykes. The estimated depth is about 0.6 times the actual depth to the top of thin dykes, and around the true depth for thick dykes having width-to-depth ratio around 3. The depth is considerably overestimated for very thick dykes (e.g., contacts, which is a special case of the thick dyke). Thus, the autocorrelation method requires that the width-to-depth ratio of the dyke is estimated independently to correctly estimate the depths. Alternatively, it must be assumed that the width-to-depth ratio for the two-dimensional source body is between 1.5 and 4.     

Early detection of large eruptions at Piton de La Fournaise volcano (La Réunion Island): Contribution of a distant tiltmeter station

We report a compilation of data recorded at a distant tiltmeter station (RER) during recent episodes of dyke emplacement and eruption (2003–2007) at Piton de La Fournaise volcano (La Réunion Island). This sensitive station provides useful information for evaluating the extent of deformation. Distinct responses of this station were recorded based on the eruption type. Dykes feeding summit eruptions did not significantly influence the RER tiltmeter signals, whereas dykes feeding large distal eruptions (with vents located more than 4 km from the summit) generated up to 1.4 μrad of tilt, an amplitude 2 to 4 times greater than for proximal eruptions (0.3–0.7 μrad) on the flanks of the summit cone. The distinct tilt amplitude is directly linked to the location, depth, and volume of the dyke. Comparison with summit tiltmeters reveals that up to one-third to half of the RER tilt signal associated to dyke propagation is recorded when the dyke is still below the summit crater. Thus, before large distal eruptions, more than 0.5 μrad of tilt is recorded in less than 20 min when the dyke is below the summit crater (i.e. a few minutes/hours before the beginning of the eruption). We can thus propose for the RER station a threshold value of 0.5 μrad which, when reached as a dyke rises beneath the summit crater, suggests a high likelihood of a large distal eruption. The distant RER tiltmeter station thus appears to be a powerful tool for forecasting the type of eruption that is likely to occur, and can contribute to the early detection of large distal eruptions at Piton de La Fournaise, which are the most dangerous to inhabitants. For volcano monitoring, installation of high precision distant tiltmeters along the lower slopes of a volcano may provide warnings of large eruptions with enough lead time to allow for short-term hazards mitigation efforts.     

Applying the tilt‐depth and contact‐depth methods to the magnetic anomalies of thin dykes

A new method for the calculation of the depth, location, and dip of thin dykes from pole‐reduced magnetic data is introduced. The depth can be obtained by measuring the distance between chosen values of a tilt angle that is based upon the ratio of the magnetic field and its Hilbert transform over the dyke. Alternatively, it can be obtained from the horizontal derivative of the ratio of the Hilbert transform of the field to the field itself, over the dyke. The latter method also allows the dip of the dyke to be estimated from the gradient of the depth estimates.     

Detection of Magnetically Susceptible Dyke Swarms in a Fresh Coastal Aquifer

Groundwater constitutes the main source of freshwater in Shalatein, on the western coast of the Red Sea, in Egypt. The fresh aquifer of Shalatein is intensively dissected by shallow and deep faults associated with the occurrence of dykes and/or dyke swarms. In this context, synthesis of electrical resistivity, ground magnetics, and borehole data was implemented to investigate the freshwater aquifer condition, locate the intrusive dykes and/or dyke swarms, and demarcate the potential freshwater zones. Nine Schlumberger VES’s with maximum current electrode half-spacing (AB/2) of 682 m were conducted. The subsurface was successfully delineated by general four layers. The fresh aquifer of the Quaternary and Pre-Quaternary alluvium sediments was effectively demarcated with true resistivities ranged from 30 to 105 Ωm and thickness ranged between 20 and 60 m. A ground magnetic survey comprised 35 magnetic profiles, each 7 km in length. Magnetic data interpretation of the vertical derivatives (first and second order), downward continuation (100 m), apparent susceptibility (depth of 100 m), and wavelength filters (Butterworth high-pass of wavelengths <100 m and Band-Pass of wavelengths 30–100 m) successfully distinguished the near surface structure with five major clusters of dyke swarms, whereas filters of the upward continuation (300 m) and Butterworth low-pass (wavelengths >300 m) clearly reflected the deep-seated structure. The computed depth by the 3D Euler deconvolution for geological contacts and faults (SI = 0) ranged from 14 to 545 m, whereas for dyke and sill (SI = 1), it ranged from 10 to 1,095 m. The western part of the study area is recommended as a potential freshwater zone as it is characterized by depths >100 m to the top of the dykes, higher thickness of the fresh aquifer (45–60 m), depths to the top of the fresh aquifer ranging from 25 to 40 m, and higher resistivities reflecting better freshwater quality (70–105 Ωm).     

Basement control on dyke distribution in Large Igneous Provinces: Case study of the Karoo triple junction

Continental flood basalts consist of vast quantities of lava, sills and giant dyke swarms that are associated with continental break-up. The commonly radiating geometry of dyke swarms in these provinces is generally interpreted as the result of the stress regime that affected the lithosphere during the initial stage of continental break-up or as the result of plume impact. On the other hand, structures in the basement may also control dyke orientations, though such control has not previously been documented. In order to test the role of pre-dyke structures, we investigated four major putative Karoo-aged dyke swarms that taken together represent a giant radiating dyke swarm (the so-called “triple-junction”) ascribed to the Jurassic Karoo continental flood basalt (> 3 × 10⁶ km²; southern Africa). One of the best tests to discriminate between neoformed and inherited dyke orientation is to detect Precambrian dykes in the Jurassic swarms. Accordingly, we efficiently distinguished between Jurassic and Precambrian dykes using abbreviated low resolution, ⁴⁰Ar/³⁹Ar incremental heating schedules.Save-Limpopo dyke swarm samples (n = 19) yield either apparent Proterozoic (728–1683 Ma) or Mesozoic (131–179 Ma) integrated ages; the Olifants River swarm (n = 20) includes only Proterozoic (851–1731 Ma) and Archaean (2470–2872 Ma) dykes. The single age obtained on one N–S striking dyke (1464 Ma) suggests that the Lebombo dyke swarm includes Proterozoic dykes in the basement as well. These dates demonstrate the existence of pre-Karoo dykes in these swarms as previously hypothesized without supporting age data. In addition, aeromagnetic and air-photo interpretations indicate that: (1) dyke emplacement was largely controlled by major discontinuities such as the Zimbabwe and Kaapvaal craton boundaries, the orientation of the Limpopo mobile belt, and other pre-dyke structures including shear zones and (2) considering its polygenetic, pre-Mesozoic origin, the Olif ants River dyke swarm cannot be considered part of the Karoo magmatic event.This study, along with previous results obtained on the Okavango dyke swarm, shows that the apparent “triple junction” formed by radiating dyke swarms is not a Jurassic structure; rather, it reflects weakened lithospheric pathways that have controlled dyke orientations over hundreds of millions of years. One consequence is that the “triple-junction” geometry can no longer be unambiguously used as a mantle plume marker as previously proposed, although it does not preclude the possible existence of a mantle plume. More generally, we suggest that most Phanerozoic dyke swarms (including triple junctions) related to continental flood basalts were probably controlled in part by pre-existing lithospheric discontinuities.     

A felsic dyke swarm formed under the sea: the Archean Hunter Mine Group,south-central Abitibi Belt,Quebec, Canada

An extensive rhyolitic dyke swarm has intruded subaqueous pyroclastic deposits, iron-formations, hyaloclastite breccias and lava flows of the 2730 Ma Hunter Mine Group (HMG) in the south-central part of the Archean Abitibi belt, Quebec. The dyke swarm has a minimum width of 500 m and can be traced perpendicular to the section for 2.4 km. Based on crosscutting relationships, chilled margins, quartz content and colour, five distinct dyke generations have been established. Each dyke generation has several magmatic pulses as indicated by parallel rows of columnar joints. Absence of brecciation between parallel rows suggests extremely brief intervals between magma pulses. The central parts of most dykes display inverted V-shaped patterns of columnar-joint convergence, inferred to indicate differential cooling during the late stages of dyke propagation. The dykes commonly display delicate spherulites suggesting rapid cooling, solidification temperatures between 400 and 600°C and penecontemporaneous devitrification. Quartz-feldspar aggregates in the groundmass have locally developed microgranophyric textures. Large spherulites near the chilled margins probably formed at temperatures below 400°C. Percolation of abundant water throughout the dyke complex is suggested by ubiquitous prominent chilled dyke margins. Development of a chilled margin 500 m along one dyke suggests that water percolated at least 500 m below the water/rock interface. Because the dykes intruded subaqueous pyroclastic deposits of similar composition, dyke emplacement below the sea floor is inferred. Interstratification of pillowed flows and brecciated pillowed flows containing rhyolite fragments at the top of the 4–5-km-thick sequence indicates that the central felsic complex probably never emerged during its evolutionary history, supporting the contention that the felsic dyke complex was emplaced beneath the Archean sea floor.     

Monitoring of suspended sediment variation using Landsat and MODIS in the Saemangeum coastal area of Korea

Suspended sediment concentration (SS) is an important indicator of marine environmental changes due to natural causes such as tides, tidal currents, and river discharges, as well as human activities such as construction in coastal regions. In the Saemangeum area on the west coast of Korea, construction of a huge tidal dyke for land reclamation has strongly influenced the coastal environment. This study used remotely sensed data to analyze the SS changes in coastal waters caused by the dyke construction. Landsat and MODIS satellite images were used for the spatial analysis of finer patterns and for the detailed temporal analysis, respectively. Forty Landsat scenes and 105 monthly composite MODIS images observed during 1985-2010 were employed, and four field campaigns (from 2005 to 2006) were performed to verify the image-derived SS. The results of the satellite data analyses showed that the seawater was clear before the dyke construction, with SS values lower than 20 g/m(3). These values increased continuously as the dyke construction progressed. The maximum SS values appeared just before completion of the fourth dyke. Values decreased to below 5 g/m(3) after dyke construction. These changes indicated tidal current modification. Some eddies and plumes were observed in the images generated from Landsat data. Landsat and MODIS can reveal that coastal water turbidity was greatly reduced after completion of the construction.     

Magma expansion and fragmentation in a propagating dyke

The influence of magma expansion due to volatile exsolution and gas dilation on dyke propagation is studied using a new numerical code. Many natural magmas contain sufficient amounts of volatiles for fragmentation to occur well below Earth's surface. Magma fragmentation has been studied for volcanic flows through open conduits but it should also occur within dykes that rise towards Earth's surface. The characteristics of volatile-rich magma flow within a hydraulic fracture are studied numerically. The mixture of melt and gas is treated as a compressible viscous fluid below the fragmentation level and as a gas phase carrying melt droplets above it. The numerical code solves for elastic deformation of host rocks, the flow of the magmatic mixture and fracturing at the dyke tip. With volatile-free magma, a dyke fed at a constant rate in a uniform medium adopts a constant shape and width and rises at a constant velocity. With volatiles involved, magma expands and hence the volume flux of magma increases. With no fragmentation, this enhanced flux leads to acceleration and thinning of the dyke. Simple scaling laws allow accurate predictions of dyke width and ascent rate for a wide range of conditions. With fragmentation, dyke behaviour is markedly different. Due to the sharp drop of head loss that occurs in gas-rich fragmented material, large internal overpressures develop below the dyke tip and induce swelling of the nose region, leading to deceleration of the dyke. These results are applied to the two-month long period of volcanic unrest that preceded the May 1980 eruption of Mount St Helens. An initial phase of rapid earthquake migration from the 7–8 km deep reservoir to shallow levels was followed by very slow progression of magma within the edifice. Such behaviour can be accounted for by magma fragmentation at the top of a dyke.     

The dynamics of magma withdrawal from a density stratified dyke

Eruptions from the top of a dyke containing two layers of magma can selectively withdraw the upper layer, leaving the dense lower layer undisturbed. Alternatively, if the upper layer is thinner than some critical depth, d, then both layers will be tapped simultaneously. Laboratory experiments yield an equation giving the draw-up depth, d, as a function of dyke geometry, eruption rate, and magma properties. This equation is valid for low to moderate Reynolds numbers and applies to dykes which are much longer than the draw-up depth. Short dykes will yield larger draw-up depths than are predicted by the equation. A large draw-up depth is favoured when the eruption rate, upper layer magma viscosity, or dyke length/breadth ratio is large or the density difference is small. Calculations show that rhyolite-capped dykes can contain several hundred metres thickness of rhyolite when a lower layer is first tapped. Draw-up depths in a dyke are as much as an order of magnitude greater than those for an identical eruption from a large cylindrical chamber tapped by a central vent. Nonetheless, for low effusion rate eruptions from small dykes, as at Inyo Domes, California, relatively small draw-up heights are calculated (e.g. 70 m). This is compatible with the small amounts of mixed magmas found at the transition between the two rhyolite magmas erupted there [11].     

基于辛算法模拟探地雷达在复杂地电模型中的传播

近年来,探地雷达(GPR)凭借其快速、高效、无破损等特点,已经广泛应用于浅地层目标探测中.数值模拟是研究探地雷达电磁波在地下结构中传播规律的有效手段.辛算法是一种保持Hamilton系统总能量不变的时域数值计算方法.本文提出了基于一阶显式辛分块龙格库塔方法的探地雷达数值模拟方法.通过对比本文算法与时域有限差分方法计算结果可知,在同等计算精度下,本文算法可以节省25%的计算时间.并基于本文算法对两个复杂GPR模型进行正演模拟,得到模拟GPR探测wiggle图,这有助于更好的理解和分析实测雷达数据.     

Modeling of spatially correlated,site-reflected,and nonstationary ground motions compatible with response spectrum

Seismic risk analysis and mitigation of spatially extended structures require the synthesis of spatially varying ground motions in the response history analysis of these structures. These synthetic motions are usually desired to be spatially correlated, site reflected, nonstationary, and compatible with target design response spectra. In this paper, a method is presented for simulating spatially varying ground motions considering the nonstationarity, local site effects, and compatibility of response spectra. The scheme for generating spatially varying and response spectra compatible ground motions is first established for spatial locations on the ground surface with varying site conditions. The design response spectrum is introduced as the “power” spectrum at the base rock. The site amplification approach is then derived based on the deterministic wave propagation theory, by assuming that the base rock motions consist of out-of-plane SH wave or in-plane combined P and SV waves propagating into the site with assumed incident angles, from which tri-directional spatial ground motions can be generated. The phase difference spectrum is employed to model ground motions exhibiting nonstationarity in both frequency and time domains with different site conditions. The proposed scheme is demonstrated with numerical examples.     

Variations in magnetic anisotropy and opaque mineralogy along a kilometer deep profile within a vertical dyke of the syenogranite porphyry at Cínovec (Czech Republic)

Vertical variations in magnetic fabric and paragenesis of the ferrimagnetic minerals for a virtually upright dyke of the Altenberg syenogranite porphyry were investigated using profile data of borehole E-16 to a depth of 922.7 m (Eastern Kru né hory Mts., NW Bohemia). It was revealed that this dyke likely consists of two magma pulses indicated both by magnetic fabric and by opaque mineralogy. In most of the profile, the magnetite grains are oriented parallel to the sub-vertical dyke. Only at the base of the upper magma pulse, in a depth interval of 200–400 m, are these grains oriented sub-horizontally being thus perpendicular to the dyke walls. This pattern can be interpreted as a consequence of a static vertical compaction of the magma of the basal portions of the upper pulse due to the pressure of the ascending lower pulse. The large planes of the ferrimagnetic minerals are oriented perpendicular to the shortening direction.